勷勤数学•专家报告

题      目：On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions

报  告  人：何凌冰 教授  (邀请人：李颖花)

                                 清华大学

时      间：12月20日  15：00-16：00

地     点：数科院西楼111报告厅

报告人简介:

       何凌冰，教授，清华大学数学系，研究领域为偏微分方程，包括流体力学方程组（Navier-Stokes, Euler,MHD），动理学方程（Boltzmann，Landau，Vlasov-Possion）。已在“Ann. Sci.Éc. Norm. Supér.”、“Ann. PDE”、“Comm. Math. Phys.”、“Arch. Ration. Mech. Anal.”、“Math. Ann.”等国际主流数学杂志发表多篇学术论文。

摘      要：

       We provide comprehensive validation of Ludwig Boltzmann’s hypotheses, as articulated in his seminal work in 1867, within the framework of the Landau equation in the presence of a harmonic potential. Our study verifies that: (i) Each {\it entropy-invariant solution} corresponds to a {\it time-periodic Maxwell-Boltzmann distribution}. Additionally, these distributions are governed by thirteen conservation laws. (ii) Each {\it time-periodic Maxwell-Boltzmann distribution} demonstrates nonlinear stability, including neutral asymptotic stability and Lyapunov stability. The rate of convergence is intricately connected to these thirteen conservation laws and is optimal in comparison to the linear case. Significantly, these results suggest that the {\it time-periodic Maxwell-Boltzmann distribution} $M$ acts as the “asymptotic state” for solutions to the equation that share the same conservation laws as $M$.

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